From 1986 to 1988 he spent some periods at the University of Aachen studying some problems in computational group theory underthe supervision of Prof. Neubuser. He got his PhD in mathematics in 1990; from 1992 to 2001 he was Associated Professor and from 2001 he is Full Professor at the University of Brescia. In 2003-2004 he was the scientific head of the research group of Brescia within the national project "Group Theory and Applications" and in 2006-2007 he has been the Scientific Coordinator of the national research "Group Theory and Applications". He works as a referee for several internationally reputed journals; moreover he is reviewer for Math. Rev. He has been invited speaker in several international conferences and in many universities both in Italy and abroad. In addition he has been scientific organizer of several international conferences.
He has written more than 60 papers, all of them in group theory. In particular he is interested in the following topics:
a) Problems concerning the minimal number of generators in a finite group.
In this context he proved for example that a finite group where all the Sylow subgroups are d-generated can be generated by d+1 elements. Moreover, he found the minimal number of generators needed for the almost simple groups and this allowed him to deduce that an almost simple group has zero presentation rank, as it was conjectured by Gruenberg. In addition, he studied the structure of groups with the following property: every proper quotient can be generated by less elements than those needed to generate the whole group. By using this result he proved that the product of 20 copies of Alt(5) is the smallest group with non zero presentation rank. Besides he found bounds to the minimal number of generators for a group which can be generated by subgroups of coprime order; these results have been used to solve an open problem concerning the possibility of generalizing the Grushko-Neumann Theorem to profinite groups.
b) Asymptotic results for permutation groups and linear groups.
He has found estimates depending on the degree of both permutation groups and linear groups to the following invariants: the minimal number of generators, the maximal rank of the abelian sections and the number of the abelian and non abelian chief factors. These results should be seen as generalizations of theorems proved by other authors (Bryant, Kovacs, Newman, Robinson, Pyber) in the solvable case. As an application we mention that an estimate of the number of transitive subgroups of Sym(n) has been obtained; furthermore, the probability that a randomly chosen subgroup of Sym(n) is transitive tends to zero as n goes to infinity and this validates a conjecture due to Erdos.
c) Probabilistic methods in group theory.
He has studied the asymptotic behaviour of the probability of generating a finite group G with d elements and he proved a conjecture due to Pak concerning the number of elements that should be considered in order to generate the group with good probability. One more result should be mentioned in this context: if a d-generated group G has a unique minimal normal subgroup N and the order of N is large enough, then given d elements which generate G modulo N we get that they generate almost certainly G. Moreover, he proved that if n is large enough and d>n/2, then in a permutation group of degree n we get that d randomly chosen elements generate almost certainly the group G.
d) Representation of finite lattices as intervals in subgroup lattice of a finite group. He made significant contributions to the study of which finite lattices can be represented as intervals in the subgroup lattice of a finite group. In particular he focused on lattices Mn which consist of a maximum, a minimum and of n pairwise incomparable elements; he built infinite families of counterexamples to the conjecture that Mn can be represented as interval in the subgroup lattice of a finite group only when n-1 is a prime power. In addition, together with Baddeley he reduced the study of this problem to issues concerning almost simple groups.
e) Permutations groups with cyclic stabilizers.
He proved that a transitive permutation group of degree n with cyclic point-stabilizers has order at most n(n-1). Furthermore, this result was generalized in two directions. In collaboration with Mainardis and Stellmacher, he classified the transitive permutation groups of degree n in which the point-stabilizers have orders precisely n-1. Besides, in a joint work with Herzog and Kaplan he proved that if A is a non trivial subgroup of a finite group G and there exists an element in the centralizer of A with order greater or equal than the index of A in G, then A contains a non trivial subgroup which is normal in G. As a consequence if A is a point-stabilizer in a transitive permutation group of degree n>1, then the center of A has exponent less than n.
More recently, his research work lead to new results in the following topics: Hurwitz groups, Carter subgroups, the probabilistic zeta function, finitely generated profinite groups, existence and number of complements of the socle in monolithic groups.